Mathematics – Functional Analysis
Scientific paper
2008-09-06
Mathematics
Functional Analysis
3 pages
Scientific paper
We characterize the projectors $ P $ on a Banach space $ E $ with the property of being connected to all the others projectors obtained as a conjugation of $ P $. Such property is described in terms of the general linear group of the spaces $ \mathrm{Range} P $, $ \mathrm{ker} P $ and $ E $. Using this characterization we show an example of Banach space where the conjugacy class of a projector splits into several arcwise components. An example of Banach algebra where the conjugacy class is greater than a connected component was shown by G. Porta and L. Recht (Acta Cientifica Venezolana, 1987) for the Banach algebra of continuous 2x2 complex matrices-valued functions on $ S^3 $. By recalling the fact that projectors with finite-dimensional ranges are connected if and only if they have the same rank, we show that an invertible operator can be connected to a direct sum $ A \oplus T $ of invertible operators on a splitting of a finite dimensional space and a finite co-dimensional complement.
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